If I have an open subset $U \subset \mathbb{R}^n$ and a distribution $\rho \in \mathscr{D}'(U; \mathbb{R})$, i.e. a continuous linear functional $\rho: \mathsf{C}_{\mathsf{c}}^\infty(U;\mathbb{R}) \rightarrow \mathbb{R}$, then I know that $\rho$ can be restricted to an open subset $V \subset U$, say to a distribution $\rho|_V \in \mathscr{D}'(V;\mathbb{R})$, in the obvious way – we can just extend any test function $\varphi \in \mathsf{C}_{\mathsf{c}}^\infty(V;\mathbb{R})$ by zero to get a test function $\bar{\varphi} \in \mathsf{C}_{\mathsf{c}}^\infty(U;\mathbb{R})$, and then set $$ \rho|_V(\varphi) := \rho(\bar{\varphi}). $$ My question is, can we do something like this if $V \subset U$ isn't open? Since if $V \subset U$ is, for example, a submanifold of smaller dimension, then the extension of a nonzero test function by zero is no longer smooth (or even continuous). The specific example I have in mind is this:

Say $\rho \in \mathscr{D}'(\mathbb{R} \times U;\mathbb{R})$ is a distribution on $\mathbb{R} \times U$, thought of as a time-dependent electric charge density on $U \subset \mathbb{R}^3$. Symbolically let's write $$ \langle \rho,\psi \rangle_{\mathbb{R} \times U} = \int_{U}\int_\mathbb{R} \rho(t,x)\psi(t,x)\:\mathsf{d}t\mathsf{d}x, $$ where the "function" $\rho(t,x): \mathbb{R} \times U \rightarrow \mathbb{R}$ doesn't actually exist unless $\rho$ is regular. Then for each time $t_0 \in \mathbb{R}$, I should intuitively have a charge density $\rho_{t_0}$ on $U$, i.e. a distribution $\rho_{t_0} \in \mathscr{D}'(U; \mathbb{R})$, obtained by somehow restricting $\rho$ to the codimension-$1$ submanifold $\{t_0\} \times U \subset \mathbb{R} \times U$; this would be given symbolically by $$ \langle \rho_{t_0}, \varphi \rangle_U = \int_U \rho(t_0,x)\varphi(x)\:\mathsf{d}x. $$ It seems like this equation can be taken as the definition of $\rho_{t_0}$ and makes complete sense, as long as $\rho$ is regular. But if $\rho$ is an arbitrary distribution, then how can I actually define $\rho_{t_0}$? Here are my ideas, which seem to have some issues:

  1. Intuitively, it seems like I should define $\langle \rho_{t_0},\varphi \rangle_U := \langle \rho\delta_{t_0}, \tilde{\varphi} \rangle_{\mathbb{R} \times U}$, where $\delta_{t_0} \in \mathscr{D}'(\mathbb{R};\mathbb{R})$ is the Dirac delta distribution at $t_0$ and $\tilde{\varphi}: \mathbb{R} \times U \rightarrow \mathbb{R}$ is the function given by $\tilde{\varphi}(t,x) := \varphi(x)$. Symbolically we would have $$ \langle \rho_{t_0},\varphi \rangle_U = \langle \rho\delta_{t_0}, \tilde{\varphi} \rangle_{\mathbb{R} \times U} = \int_U\int_{\mathbb{R}} \rho(t,x)\delta_{t_0}(t)\tilde{\varphi}(t,x)\:\mathsf{d}t\mathsf{d}x = \int_U \rho(t_0,x)\varphi(x)\:\mathsf{d}x, $$ which is exactly what it should be. The problem is that I need to multiply the two distributions $\rho$ and $\delta_{t_0}$ together (plus, one is distribution on $\mathbb{R} \times U$ and the other is a distribution on $\mathbb{R}$).

  2. Instead of letting $\rho$ be a distribution on $\mathbb{R} \times U$, I could take it to be a parametrized family of distributions on $U$, say $\rho: \mathbb{R} \rightarrow \mathscr{D}'(U;\mathbb{R})$, given by $t \mapsto \rho_t$. The problem here is that I no longer have a notion of $\frac{\partial \rho}{\partial t}$, and even if I did, then it's hard to make sense of something like $\frac{\partial^2 \rho}{\partial t \partial x_1}$.

  3. Lastly I could try something like this: to define the value of $\rho_{t_0}$ on some test function $\varphi \in \mathsf{C}_{\mathsf{c}}^\infty(U;\mathbb{R})$, take some sequence $\varphi_1,\varphi_2,\dots \in \mathsf{C}_{\mathsf{c}}^\infty(\mathbb{R} \times U;\mathbb{R})$ of test functions whose supports form a decreasing chain approaching $\{t_0\} \times U$, i.e. $\bigcap_{j=1}^\infty \mathsf{supp}(\varphi_j) = \{t_0\} \times U$. Then define $$ \langle \rho_{t_0},\varphi \rangle_U := \lim_{j \rightarrow \infty} \langle \rho,\varphi_j \rangle_{\mathbb{R} \times U}. $$ This seems kind of complicated, and there may be some issues with well definedness.

Is this notion written down anywhere, in any reference? Any help is really appreciated!

  • 1
    $\begingroup$ good question. the canonical approach is 3 which depending on the situation may or may not succeed. What you seem to be interested in is restricting a distribution to an affine subspace. Results in the literature which do that are generically called "trace theorems" (google that). You can also use Hormander's notiion of wavefront set of a distribution. Modulo good geometric position relative to the subspace there are theorems which guarantee that method 3 will work. See Corollary 8.2.7 in Hormander vol 1, second edition. $\endgroup$ Feb 27, 2020 at 14:39
  • $\begingroup$ That's exactly what I was looking for, thank you so much! I guess I must have missed that in Hormander's book. $\endgroup$
    – Alec
    Feb 27, 2020 at 18:14
  • $\begingroup$ If you are already reading Hormander, then you are on the right track! A remark about trace theorem versus wavefront: if you are only interested in the restriction of one distribution then probably the wavefront theory will be more helpful, if you want to do that in bulk for a lot of distributions at once, you may need a trace thm which typically needs a regularity hypothesis like being in a Sobolev with high enough exponent. See, e.g., these notes I found on the web ltcc.ac.uk/media/london-taught-course-centre/documents/… $\endgroup$ Feb 27, 2020 at 18:36
  • $\begingroup$ I haven't really read into wavefronts, so I'll take a look at that – thanks again for your suggestions! $\endgroup$
    – Alec
    Feb 27, 2020 at 19:46
  • $\begingroup$ check this out also arxiv.org/abs/1404.1778 $\endgroup$ Feb 28, 2020 at 0:15


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